Spectral Exchange of Period and L-Sheaves

Definitions and mathematical statement

The supplied record identifies this question as Conjecture 1, Global Geometric Duality, citing [BZSV24], Conjecture 12.1.1. It uses the notation $\mathcal P_{X,\Psi}$ for a period sheaf and $\mathcal L_{X^\vee,\Psi^\vee}$ for an $L$-sheaf associated with dual data. Write $\operatorname{spec}$ for the spectral projection mentioned in the source. These symbols name the source's intended objects; their constructions and domains are not supplied.

Conjectural statement, conditional on the missing setup. For the data and under the hypotheses intended in [BZSV24], Conjecture 12.1.1, the geometric Langlands correspondence identifies the spectral projection of the period sheaf with the corresponding $L$-sheaf. Schematically, \[ \operatorname{spec}\left(\mathcal P_{X,\Psi}\right) \quad\longleftrightarrow\quad \mathcal L_{X^\vee,\Psi^\vee}. \] Here the arrow denotes the identification asserted under the intended correspondence, not a specified morphism between objects in a common category.

Unresolved qualifications. The supplied diagram is corrupted. Although it contains the expressions $\operatorname{AUT}(\operatorname{Bun}_G)$ and $\operatorname{QC}^{!}(\operatorname{Loc}_{G^\vee})$, it does not reliably specify their definitions, the direction or normalization of the correspondence, or the placement of the projection. The base field, underlying curve, allowed groups $G$, admissible data $(X,\Psi)$, meaning of their duals, sheaf theory, and precise categorical identification are absent. Consequently, the supplied material does not determine a fully quantified, self-contained mathematical formulation; the displayed assertion preserves only its stated target.

Short English statement

The conjecture asks whether geometric Langlands carries the spectral projection of the period sheaf to the $L$-sheaf for the dual data. The supplied fragment omits the definitions and hypotheses needed to make this a precise categorical assertion, so this is a schematic statement requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30006324$, title Spectral Exchange of Period and L-Sheaves, exact source problem code catalogue entry. The record gives proposed year 2025 and no named proposer.
• Arbeitsgemeinschaft: Relative Langlands Duality (2025), Oberwolfach Reports, pp. 811--882, DOI 10.4171/owr/2025/18. Supplied report and literature-source link: https://doi.org/10.4171/owr/2025/18. The extracted passage labels the assertion Conjecture 1 (Global Geometric Duality) and attributes it to [BZSV24], Conjecture 12.1.1; a full bibliographic expansion of that label is not supplied.
• Supplied literature source: Relative Langlands Duality, https://arxiv.org/abs/2409.04677.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and says that the spectral exchange remains conjectural beyond special cases. This is supplied dated curation, not an independently verified assessment; the special cases are not identified. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied registry labels the statement corrected_verified and reports that the official Oberwolfach report was checked and its notation repaired. These are source-provided verification claims; no independent source consultation or verification is asserted here.